Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866909170788204544 |
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| author | Alfaya, David Oliveira, André |
| author_facet | Alfaya, David Oliveira, André |
| contents | Let $\mathcal{L}=(L,[\cdot\,,\cdot],δ)$ be an algebraic Lie algebroid over a smooth projective curve $X$ of genus $g\geq 2$ such that $L$ is a line bundle whose degree is less than $2-2g$. Let $r$ and $d$ be coprime numbers. We prove that the motivic class of the moduli space of $\mathcal{L}$-connections of rank $r$ and degree $d$ over $X$ does not depend on the Lie algebroid structure $[\cdot\,,\cdot]$ and $δ$ of $\mathcal{L}$ and neither on the line bundle $L$ itself, but only on the degree of $L$ (and of course on $r$, $d$ and $X$). In particular it is equal to the motivic class of the moduli space of $K_X(D)$-twisted Higgs bundles of rank $r$ and degree $d$, for $D$ any effective divisor with the appropriate degree. As a consequence, similar results (actually slightly stronger) are obtained for the corresponding $E$-polynomials. Some applications of these results are then deduced. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_12246 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces Alfaya, David Oliveira, André Algebraic Geometry Let $\mathcal{L}=(L,[\cdot\,,\cdot],δ)$ be an algebraic Lie algebroid over a smooth projective curve $X$ of genus $g\geq 2$ such that $L$ is a line bundle whose degree is less than $2-2g$. Let $r$ and $d$ be coprime numbers. We prove that the motivic class of the moduli space of $\mathcal{L}$-connections of rank $r$ and degree $d$ over $X$ does not depend on the Lie algebroid structure $[\cdot\,,\cdot]$ and $δ$ of $\mathcal{L}$ and neither on the line bundle $L$ itself, but only on the degree of $L$ (and of course on $r$, $d$ and $X$). In particular it is equal to the motivic class of the moduli space of $K_X(D)$-twisted Higgs bundles of rank $r$ and degree $d$, for $D$ any effective divisor with the appropriate degree. As a consequence, similar results (actually slightly stronger) are obtained for the corresponding $E$-polynomials. Some applications of these results are then deduced. |
| title | Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2102.12246 |